Can I run a GLMM model when I have one observation for most subjects? 
I have a binary DV and my panel data set contains more than one observation
for only 20% of the subjects which makes it very unbalanced.
Is there anything methodologically wrong with doing a mixed model with data that looks like this,i.e. with many subjects having only one observation? 
I'd appreciate any hints.

No. of observations per subject:       1       2           3       4       5    
  Frequency:                                 4145     821    176     57     19

I am reading "Linear Mixed Models A Practical Guide Using Statistical Software" by West, Welch and Galecki, and my data set does not seem to exactly fit into their description of three types of data they discuss for mixed models, namely Clustered Data, Repeated-Measures and Longitudinal Data. 
 A: In section 12.9 of Gelman and Hill's book, namely "Data Analysis Using Regression Analysis and Multilevel/Hierarchical Models" we read:
"
How many observations per group?
Even two observations per group is enough to fit a multilevel model. It is even
acceptable to have one observation in many of the groups. When groups have few
observations, their αj ’s won’t be estimated precisely, but they can still provide partial information that allows estimation of the coefficients and variance parameters
of the individual- and group-level regressions."
A: This depends on what you know about the science behind the data.  A mixed model will assume that the structure of all the unobserved pieces (variability within subjects with only 1 obsevation, etc.) is exactly the same as those from which you have more data.  Is this an assumption that you consider to be reasonable?  The data cannot tell you if this is an assumption you can live with, this has to come from your knowledge of the underlying science.
The power (ability of any test to tell you meaningful results) for a binary response variable is more dependent on the smaller of the number of successes and number of failures than the overall sample size, so unless the mean proportion is close to 0.5 in your case you are unlikely to have much power to find anything meaningful with at most 5 observations from a single subject.
If you can find some meaningful prior information then a Bayesian approach with informative prior (based on background and science, not the current data) may give you a better chance of getting meaningful answers.  Otherwise you will probably need to bound the variance on the random effects to be fairly small to get anything meaningful.
